Statistica Sinica 12(2002), 1061-1072

EMPIRICAL BAYES TESTS BASED ON

KERNEL SEQUENCE ESTIMATION

Jianjun Li and Shanti S. Gupta

Purdue Univeristy

Abstract: In this paper, we consider the hypothesis-testing problem in the continuous one-parameter exponential family using the nonparametric empirical Bayes approach. In order to estimate an unknown marginal density and its derivative, a kernel sequence method is introduced. This method uses a sequence of kernel functions and allows the kernel index and window bandwidth to vary simultaneously. Thus improved estimates are obtained. Then we construct a monotone empirical Bayes test based on these estimates and show that the rule has a rate of convergence of $(\ln n)^{3+\epsilon}/n$ for any $\epsilon>0$. This rate substantially improves the previous results and is much closer to the lower bound rate $1/n$. Since the rule is monotone, it also has good performance for small samples.

Key words and phrases: Empirical bayes, kernel sequence method, rate of convergence, regret bayes risk.